<s>
In	O
geometry	O
,	O
a	O
centroidal	B-Application
Voronoi	I-Application
tessellation	I-Application
(	O
CVT	O
)	O
is	O
a	O
special	O
type	O
of	O
Voronoi	B-Architecture
tessellation	I-Architecture
in	O
which	O
the	O
generating	O
point	O
of	O
each	O
Voronoi	B-Architecture
cell	I-Architecture
is	O
also	O
its	O
centroid	O
(	O
center	O
of	O
mass	O
)	O
.	O
</s>
<s>
A	O
number	O
of	O
algorithms	O
can	O
be	O
used	O
to	O
generate	O
centroidal	B-Application
Voronoi	I-Application
tessellations	I-Application
,	O
including	O
Lloyd	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
K-means	B-Algorithm
clustering	I-Algorithm
or	O
Quasi-Newton	B-Algorithm
methods	I-Algorithm
like	O
BFGS	B-Algorithm
.	O
</s>
<s>
Gersho	O
's	O
conjecture	O
,	O
proven	O
for	O
one	O
and	O
two	O
dimensions	O
,	O
says	O
that	O
"	O
asymptotically	O
speaking	O
,	O
all	O
cells	O
of	O
the	O
optimal	O
CVT	O
,	O
while	O
forming	O
a	O
tessellation	B-Algorithm
,	O
are	O
congruent	O
to	O
a	O
basic	O
cell	O
which	O
depends	O
on	O
the	O
dimension.	O
"	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
the	O
basic	O
cell	O
for	O
the	O
optimal	O
CVT	O
is	O
a	O
regular	O
hexagon	B-Application
as	O
it	O
is	O
proven	O
to	O
be	O
the	O
most	O
dense	O
packing	O
of	O
circles	O
in	O
2D	O
Euclidean	O
space	O
.	O
</s>
<s>
Centroidal	B-Application
Voronoi	I-Application
tessellations	I-Application
are	O
useful	O
in	O
data	B-General_Concept
compression	I-General_Concept
,	O
optimal	O
quadrature	B-Algorithm
,	O
optimal	O
quantization	B-Algorithm
,	O
clustering	B-Algorithm
,	O
and	O
optimal	O
mesh	O
generation	O
.	O
</s>
<s>
A	O
weighted	O
centroidal	O
Voronoi	B-Architecture
diagrams	I-Architecture
is	O
a	O
CVT	O
in	O
which	O
each	O
centroid	O
is	O
weighted	O
according	O
to	O
a	O
certain	O
function	O
.	O
</s>
<s>
Many	O
patterns	O
seen	O
in	O
nature	O
are	O
closely	O
approximated	O
by	O
a	O
centroidal	B-Application
Voronoi	I-Application
tessellation	I-Application
.	O
</s>
